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For example, second-order arithmetic can express the principle " Every countable vector space has a basis " but it cannot express the principle " Every vector space has a basis ".

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## Some Related Sentences

For and example

__For__

**the**family is

**the**simplest

__example__of just such

**a**unit

**,**composed of people

**,**which gives us both some immunity from

**,**and

**a**way of dealing with

**,**other people.

__For__

__example__

**,**suppose

**a**man wearing

**a**$200 watch

**,**driving

**a**1959 Rolls Royce

**,**stops to ask

**a**man on

**the**sidewalk

**,**`` What time is

**it**''??

__For__

__example__

**,**there are persons who are in physical science

**,**in

**the**field of mineralogy

**,**trained in crystallography

**,**who use only X-rays

**,**applying only

**the**powder technique of X-ray diffraction

**,**to clay minerals only

**,**and who have spent

**the**last fifteen years concentrating on

**the**montmorillonites ; ;

__For__

__example__

**,**he captured some persons from York County

**,**who with teams were taking to Philadelphia

**the**furniture of

**a**man who had just been released from prison through

**the**efforts of his wife

**,**and who apparently was helpless to prevent

**the**theft of his household goods.

__For__

__example__

**,**even

**the**most successful executive lives in

**a**two-room apartment while ordinary people rent

**space**in

**the**stairwells of office buildings in which to sleep at night ; ;

__For__

__example__

**,**in

**the**third chapter of Matthew

**,**verses 13-16

**,**describing

**the**baptism of Jesus

**,**

**the**1611 version reads:

__For__

__example__

**,**

**it**probably will be necessary for

**the**Corps to have authority to pay medical expenses of volunteers.

__For__

__example__

**,**

**the**importance of

**the**Regulus 2

**,**

**a**very promising aerodynamic ship-to-surface missile designed to be launched by surfaced submarines

**,**was greatly diminished by

**the**successful acceleration of

**the**much more advanced Polaris ballistic missile launched by submerged submarines.

__For__

__example__

**,**

**the**interest of past members of

**the**Foundation's Advisory Board remains such that they place their knowledge and judgments at our disposal much as they had done when they were

**,**formally

**,**members of that Board.

__For__

__example__

**,**out of

**the**social evils of

**the**English industrial revolution came

**the**novels of Charles Dickens ; ;

__For__an

__example__let's dream up an engine that

**has**

**a**final combustion chamber volume of 5 cubic inches and

**a**cylinder volume of 45 cubic inches.

__For__

__example__

**,**

**a**Browning trap version of

**the**Superposed over/under

**,**

**the**Broadway ( from $350 up

**,**depending on grade )

**,**differs from standard models in that

**it**is equipped with

**a**full beavertail fore end

**,**

**a**cushion recoil pad and

**a**barrel-wide ventilated rib for fast sighting.

__For__

__example__

**,**

**the**Chamber of Commerce of Gatlinburg

**,**Tennessee

**,**sponsors special camera tours into

**the**Great Smoky Mountains to get pictures of

**the**profusion of wild flowers flourishing in these wooded regions.

__For__

__example__

**,**

**the**officials of Poughkeepsie town ( township ) where

**the**project is located think highly of

**it**because

**it**simplifies their snow clearing problem.

__For__

__example__

**,**don't pay in

**a**truck policy for medical coverage that you may be paying for in

**a**health and accident policy.

__For__

__example__: If your bodily injury claims start payment after

**the**first $250

**,**

**a**25% premium saving is often made.

For and second-order

__For__one-dimensional simple harmonic motion

**,**

**the**equation of motion

**,**which is

**a**

__second-order__linear ordinary differential equation with constant coefficients

**,**could be obtained by means of Newton's second law and Hooke's law.

__For__instance

**,**retaining

**the**first two terms of

**the**series yields

**the**

__second-order__approximation to f ’( x ) mentioned at

**the**end of

**the**section Higher-order differences.

__For__

**example**

**,**

**the**sentence involving Napoleon

**can**be rewritten as “ any group of people that includes me and

**the**parents of each person in

**the**group must also include Napoleon ,” which is easily interpreted as

**a**statement in

__second-order__logic ( one would naturally start by assigning

**a**name

**,**such as G

**,**to

**the**group of people under consideration ).

__For__

**example**

**,**

**a**

__second-order__Butterworth filter will reduce

**the**signal amplitude to one fourth its original level every time

**the**frequency doubles ( so power decreases by 12 dB per octave

**,**or 40 dB per decade ).

__For__

__second-order__divisions ( under provinces and special administrative regions ), there are counties

**,**provincial cities ( 56 ), bureaus ( 34 ) and management bureaus ( 7 ).

__For__an infinitesimal deformation

**the**displacements and

**the**displacement gradients are small compared to unity

**,**i. e., and

**,**allowing for

**the**geometric linearisation of

**the**Lagrangian finite strain tensor

**,**and

**the**Eulerian finite strain tensor

**,**i. e.

**the**non-linear or

__second-order__terms of

**the**finite strain tensor

**can**be neglected.

__For__

**example**

**,**money satisfies no biological or psychological needs

**,**

**but**

**a**pay check appears to reduce drive through

__second-order__conditioning.

__For__

**example**

**,**while

**the**analysis of

**a**table ( matrix

**,**or

__second-order__arry ) of data is routine in several fields

**,**multiway methods are applied to data sets that involve 3rd

**,**4th

**,**or higher-orders.

__For__

**example**

**,**

**the**

__second-order__sentence says that for every set P of individuals and every individual x

**,**either x is in P or

**it**is not ( this is

**the**

**principle**of bivalence ).

__For__

**example**

**,**if

**the**domain is

**the**set of all real numbers

**,**one

**can**assert in first-order logic

**the**existence of an additive inverse of each real number by writing ∀ x ∃ y ( x + y = 0 )

**but**one needs

__second-order__logic to assert

**the**least-upper-bound property for sets of real numbers

**,**which states that every bounded

**,**nonempty set of real numbers

**has**

**a**supremum.

__For__

**example**

**,**

**a**

__second-order__Butterworth filter

**,**which

**has**maximally flat passband frequency response

**,**

**has**

**a**of.

__For__instance

**,**

**a**

__second-order__low-pass notch filter section only reduces ( rather than eliminates ) very high frequencies

**,**

**but**

**has**

**a**steep response falling to zero at

**a**specific frequency (

**the**so-called notch frequency ).

__For__

**a**boundary value problem of

**a**

__second-order__ordinary differential equation

**,**

**the**method is stated as follows.

__For__

**example**

**,**

**the**Löwenheim number of

__second-order__logic is already larger than

**the**first measurable cardinal

**,**if such

**a**cardinal exists.

**"**It is possible to define

**a**formula True ( n ) whose extension is T *,

**but**only by drawing on

**a**metalanguage whose expressive power goes beyond that of L.

__For__

**example**

**,**

**a**truth predicate for first-order

**arithmetic**

**can**be defined in

__second-order__

**arithmetic**.

__For__

**a**

__second-order__linear autonomous systems

**,**

**a**critical point is

**a**saddle point if

**the**characteristic equation

**has**one positive and one negative real eigenvalue.

__For__comparison

**,**in

**the**equivalent Euler – Lagrange equations of motion of Lagrangian mechanics

**,**

**the**conjugate momenta also do not appear ; however

**,**those equations are

**a**system of N

**,**generally

__second-order__equations for

**the**time evolution of

**the**generalized coordinates.

__For__

**example**

**,**PH

**,**

**the**union of all complexity classes in

**the**polynomial hierarchy

**,**is precisely

**the**class of languages expressible by statements of

__second-order__logic.

*

__For__**a**__second-order__polynomial**,****,**all**the**roots are in**the**left half plane ( and**the**system with characteristic equation is stable ) if all**the**coefficients satisfy.__For__

**example**

**,**an animal might first learn to associate

**a**bell with food ( first-order conditioning ),

**but**then learn to associate

**a**light with

**the**bell (

__second-order__conditioning ).

__For__

**example**

**,**if MK is consistent then

**it**

**has**

**a**

**countable**first-order model

**,**while

__second-order__ZFC

**has**no

**countable**models.

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